There are several definitions of fractional differential equations and
the same is true for fractional difference equations.
We will study the
definition obtained by Gejji and Deshpande [1]}. The evolution depends on
the value at all previous time-steps. The weight of previous values decays as a power-law and there is a long term memory built-in in the system.
It is not surprising that the fluctuations also decay extremely slowly
and decay can be approximated by power-law with power related to the
order of fractional difference equation. This is in turn related to the
properties of Mittag-Leffler function which plays an important role
in fractional differential equations and plays a similar role here.
In difference equations, the concept of coupled maps was introduced by
Kapral, Kuznetsov, and Kaneko. Kaneko can be credited for making it
popular.
